Concept Architecture
Concept
Theoretically, Constrained Optimisation is a mathematical framework for identifying the best feasible decision when an objective must be maximised or minimised subject to one or more restrictions. It is founded on optimisation theory and represents decision-making under scarcity, where available resources, technical requirements, policy rules or behavioural conditions limit the set of admissible choices. The concept exists because health economic decisions rarely involve unconstrained choices and must instead balance competing objectives within explicit limits.
Mathematically, constrained optimisation is represented by an objective function evaluated over a feasible set defined by equality and inequality constraints. The recognised framework uses the Lagrangian function and Karush?Kuhn?Tucker conditions to characterise optimal solutions. The associated Lagrange multipliers represent the marginal change in the optimal objective value resulting from a marginal relaxation of a binding constraint.
In practice, constrained optimisation is implemented by specifying decision variables, an objective function and the relevant constraints, then solving the resulting problem analytically or numerically. In health economics, it is used in resource allocation, formulary design, treatment selection, workforce planning, budget-constrained health maximisation and the optimisation of service configurations.
Purpose
Used to identify the allocation or decision that maximises health, welfare, utility or net benefit, or minimises cost, subject to budgetary, clinical, capacity, equity or operational constraints.
Mathematical Formulae
Primary Formula
max f(x)
subject to
g?(x) � 0, j = 1, ?, m
h?(x) = 0, k = 1, ?, p
Supporting Formulae
Lagrangian function:
L(x, ?, ?) = f(x) ? ????? ??g?(x) ? ????? ??h?(x)
Stationarity condition:
??L(x*, ?, ?) = 0
Primal feasibility:
g?(x*) � 0
h?(x*) = 0
Dual feasibility:
??* � 0
Complementary slackness:
??g?(x) = 0
Related Mathematical Methods
- Lagrange Multiplier Method
- Karush?Kuhn?Tucker Conditions
- Linear Programming
- Non-linear Programming
- Integer Programming
- Dynamic Programming
- Convex Optimisation
- Mathematical Programming
Example
A health authority has a budget of �10 million and must allocate funding between two interventions. Intervention A costs �20,000 per patient and generates 1.5 quality-adjusted life-years, while intervention B costs �10,000 per patient and generates 0.6 quality-adjusted life-years.
The optimisation problem is:
max Q = 1.5x? + 0.6x?
subject to:
20,000x? + 10,000x? � 10,000,000
x? � 0
x? � 0
Where x? and x? are the numbers of patients receiving interventions A and B. If no additional capacity constraints apply, the quality-adjusted life-year return per pound is higher for intervention A:
1.5 � 20,000 = 0.000075 QALYs per �
0.6 � 10,000 = 0.000060 QALYs per �
The constrained optimum therefore allocates the full budget to intervention A:
x? = 10,000,000 � 20,000 = 500 patients
Total health gain:
Q = 1.5 ? 500 = 750 QALYs
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B6,C2:C6) | Calculate total cost, total health gain or net benefit across funded interventions |
| SUM | =SUM(D2:D6) | Aggregate resource use or outcomes across decision variables |
| IF | =IF(E2<=$H$2,"Feasible","Constraint Violated") | Test whether a proposed allocation satisfies a budget or capacity constraint |
| MAX | =MAX(F2:F100) | Identify the highest feasible objective value among evaluated scenarios |
| Solver | Set objective cell to Max, changing B2:B6, subject to D10<=$H$2 and B2:B6>=0 | Maximise health gain or net benefit subject to budget, capacity and policy constraints |
VBA (Optional)
VBA may automate repeated constrained optimisation across multiple budgets, intervention sets and scenario assumptions using Excel Solver.
Sources
- Chiang AC, Wainwright K. Fundamental Methods of Mathematical Economics. McGraw-Hill.
- Boyd S, Vandenberghe L. Convex Optimization. Cambridge University Press.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- Stinnett AA, Paltiel AD. Mathematical Programming for the Efficient Allocation of Health Care Resources. Journal of Health Economics. 1996.
- NICE. Health Technology Evaluations: The Manual.
Related Concepts (4)
Library
Publications
1
Decision Modelling for Health Economic Evaluation — Briggs, Claxton & Sculpher, 1st Edition ed., 2006 (Oxford University Press)
Foundational textbook on decision-analytic modelling for economic evaluation, covering decision trees, Markov models, handling parameter and structural uncertainty, probabilistic sensitivity analysis, and value of information. Volume 1 in the Handbooks in Health Economic Evaluation series.
BookView source →
Frequently Asked Questions (6)
What is constrained optimisation?
A mathematical approach to finding the best outcome, such as maximum health benefit, subject to limiting conditions, such as a fixed budget.
Source: Varian 2014
What is the objective function in constrained optimisation?
A constrained optimisation problem has two parts, an objective function, the quantity to be made as large or small as possible, and one or more constraints that limit the choices allowed. In a health application the objective might be total health gain and the constraint a fixed budget, so the problem is to find the mix of services that yields the most health without exceeding the money available. Setting up the objective and constraints correctly is what turns a vague aim into a solvable problem. Weinstein and Zeckhauser (1973) apply this framing to health.
Source: Weinstein & Zeckhauser 1973
What are the elements of a constrained optimisation problem?
A constrained optimisation problem has three elements: an objective function to be maximised or minimised, such as total health benefit; decision variables that can be chosen, such as how much to spend on each use; and constraints that limit the choices, such as a fixed budget or capacity. The solution is the set of variable values that gives the best value of the objective while satisfying all the constraints. These elements formalise the problem of doing as well as possible within limits.
Source: Varian 2014
How is a constrained optimisation problem solved?
A constrained optimisation problem is solved by finding the values of the decision variables that optimise the objective subject to the constraints, often using methods such as the Lagrange multiplier technique, which incorporates the constraints into the objective. At the optimum, the objective cannot be improved without violating a constraint. The Lagrange multipliers have an economic interpretation as shadow prices, the value of relaxing a constraint slightly, such as the health gained from a marginal increase in the budget.
Source: Varian 2014
How does constrained optimisation relate to resource allocation?
Constrained optimisation relates directly to resource allocation, which is the problem of maximising benefit from limited resources. Allocating a fixed health budget to maximise health is a constrained optimisation, with health as the objective and the budget as the constraint. The solution directs resources to where they yield most benefit at the margin, equalising the marginal benefit per pound across uses. The shadow price of the budget constraint is the health obtainable from an extra pound, which grounds the cost-effectiveness threshold.
Source: Varian 2014
What are the limitations of constrained optimisation in health?
Constrained optimisation gives clear answers only when the objective, variables, and constraints can be specified, but in health these are often uncertain or contested: the objective may involve equity as well as total health, the effects of spending may be poorly known, and constraints beyond the budget may bind. The mathematics also assumes the objective can be maximised precisely, which real data rarely allow. It provides a rigorous framework for allocation but depends on inputs and value judgements that must be supplied.
Source: Varian 2014
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 29 Sep 2025
Content version: 1.0.0
Canonical Identity
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- HE-EM-DM-013
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